Advanced Simulation Methods

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Advanced Simulation Methods
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Overview

• Advanced Simulation Applications
• Queueing Models
• Extend
• HOM
• Retirement Planning
• Market Shares
• Securities Pricing
• Asian Option
• Evaluation of Hedging Strategies
• Currency Risk

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HOM Queueing Notes

• Not especially user-friendly
• Quite powerful if you learn how to use it
• Be careful entering data for the General
  arrival or service distributions
• The rate is in events per time, but the
  variance is in squared time per event

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Retirement Planning
Amanda has 30 years to save for her retirement.
At the beginning of each year, she puts 5000
into her retirement account. At any point in
time, all of Amanda's retirement funds are tied
up in the stock market. Suppose the annual return
on stocks follows a normal distribution with mean
12 and standard deviation 25. What is the
probability that at the end of 30 years, Amanda
will have reached her goal of having 1,000,000
for retirement? Assume that if Amanda reaches her
goal before 30 years, she will stop investing.
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The annual investment activities (columns A-D,
beginning in row 5) actually extend down to row
35, to include 30 years of simulated returns.
The range C6C35 will be random numbers,
generated by _at_Risk. We could track Amandas
simulated investment performance either with cell
F5 (simply D35, the final amount in Amandas
retirement account), or with F4 (the maximum
amount over 30 years). Using F4 allows us to
assume that she would stop investing if she ever
reached 1,000,000 at any time during the 30
years, which is the assumption given in the
problem statement. Cell H1 is either 1 (she made
it to 1 million) or 0 (she didnt). Over many
trials, the average of this cell will be out
estimate of the probability that Amanda does
accumulate 1 million. This will be an _at_Risk
output cell.
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It looks like Amanda has about a 48 chance of
meeting her goal of 1 million in 30 years.
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Market Shares
Suppose that each week every family in the United
States buys a gallon of orange juice from company
A, B, or C. Let PA denote the probability that
a gallon produced by company A is of
unsatisfactory quality, and define PB and PC
similarly for companies B and C.
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If the last gallon of juice purchased by a family
is satisfactory, then the next week they will
purchase a gallon of juice from the same company.
If the last gallon of juice purchased by a
family is not satisfactory, then the family will
purchase a gallon from a competitor. Consider a
week in which A families have purchased juice A,
B families have purchased juice B, and C families
have purchased juice C.
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Assume that families that switch brands during a
period are allocated to the remaining brands in a
manner that is proportional to the current market
shares of the other brands. Thus, if a customer
switches from brand A, there is probability B/(B
C) that he will switch to brand B and
probability C/(B C) that he will switch to
brand C. Suppose that 1,000,000 gallons of orange
juice are purchased each week. After a year, what
will the market share for each firm be? Assume PA
0.10, PB 0.15, and Pc 0.20.
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Now recall that a binomial random variable X is
an integer between 0 and n, viewed as the number
of successes out of n trials. The binomial
distribution assumes that there is a probability
p of a success on any one trial, and that all
trials are independent of each other. In this
case, X is the number of gallons that are bad,
n is the total number of gallons purchased of a
particular brand, and p is the probability that
any one gallon is bad.
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We can use the binomial distribution again here.
The number of people who switch from Brand A to
Brand B in any given week will be a binomial
random variable, with n equal to the total number
of people who abandon Brand A in that week, and p
equal to the proportion of the non-Brand A market
held by Brand B in that week, or B/(B C).
(Recall that the problem asks us to Assume
that families that switch brands during a period
are allocated to the remaining brands in a manner
that is proportional to the current market shares
of the other brands.)
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Well set up the ns for these binomial random
variables in columns K, L, and M, using MAX
functions as before to make sure that they never
go below 1. In column N we calculate the
proportion of non-A customers who buy B in the
current week, once again using a MAX function to
make sure this is never zero.
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The number of A customers who switch from A to
Brand C is calculated in column P it is simply
the difference between K7 and O7. We model the
switching behavior of former B customers in
columns Q, R, and S, and former C customers in
columns T, U, and V. Finally, the various
numbers of switchers are taken into account for
the start of the next week in columns B, C, and
D. All of the binomial assumptions (columns H,
I, J, O, R, and U) get copied down through row
58, so we can model a 52-week year.
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Suppose a 1 increase in market share is worth
10,000 per week to company A. Company A
believes that for a cost of 1 million per year
it can cut the percentage of unsatisfactory juice
cartons in half. Is this worthwhile? (Use the
same values of PA, PB, and PC as in part a.)
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There are a number of ways to approach this kind
of issue. One elegant way is to run two
simulations simultaneously, in which the only
difference is (in this case) the different value
for PA. Well run the same model as before, but
add to it, in parallel, a second model in which
PA 0.05 instead of 0.10. The old model is in a
worksheet called Part (a) and the new model is in
a worksheet called Part (b).
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Securities Pricing
George Brickfields business is highly exposed to
volatility in the cost of electricity. He has
asked his investment banker, Lisa Siegel, to
propose an option whereby he can hedge himself
against changes in the cost of a kilowatt hour of
electricity over the next twelve months.
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Lisa thinks that an Asian option would work
nicely for Georges situation. An Asian option
is based on the average price of a kilowatt hour
(or other underlying commodity) over a specified
time period.
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• In this case, Lisa wants to offer George a one
  year Asian option with a target price of 0.059.
• If the average price per kilowatt hour over the
  next twelve months is greater than this target
  price, then Lisa will pay George the difference.
• If the average price per kilowatt hour over the
  next twelve months is less than this target
  price, then George loses the price he paid for
  the option (but he is happy, because he ends up
  buying relatively cheap electricity).

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What is a fair price for Lisa to charge for 1
million kwh worth of these options? Use the
historical data provided and Monte Carlo
simulation to arrive at a fair price.
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Evaluation of Hedging Strategies
It is July 1, 2002, and international
entrepreneurs Clifford Kearns (CK) are
concerned about volatility in the exchange rates
between U.S. dollars and certain European
currencies. CK have incurred costs in dollars
to develop, produce, and distribute merchandise
to Norway, Switzerland, and Great Britain, for
which they expect to realize revenues in 12
months.
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Specifically, they expect to earn 1 million units
each of British pounds, Swiss francs, and
Norwegian kroner. Based on current exchange
rates, this should result in 2,337,700 in
revenue (see current rates below).
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Unfortunately, it is possible that one or more of
these currencies could devalue against the dollar
in that one year, causing CK to realize a
smaller total revenue (in dollars) than expected.
CK has turned to their investment bank, Nuccio,
Noto, and Rizzi (NNR) for advice. NNR has
recommended buying 1.3 million 1-year Euro put
options with a strike price of 0.98, for 0.0432
each. NNR claims that this hedging strategy will
substantially decrease the risk of a large loss
due to exchange rate fluctuations.
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(a) Create a simulation model to study the
unhedged distribution of revenue for CK, using
the historical exchange rate data in Exhibit 2.
Make a histogram and report summary statistics.
What is the 5 value at risk (VAR) for CKs
revenue from these three countries over the next
12 months? What is the probability that CKs
revenue will be less than 2,087,700 (i.e., a
250,000 loss or worse)? (b) Create a simulation
model to study the hedged distribution of
revenue for CK. Make a histogram and report
summary statistics with the policy recommended by
NNR. What is the 5 VAR for CKs revenue from
these three countries over the next 12 months?
What is the probability that CKs revenue will
be less than 2,087,700?
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Here are summary statistics for each of the
currencies returns against the dollar, including
a t-test to see if the means are significantly
different from zero (they are not)
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It turns out that all four of our variables can
be modeled reasonably well by normal
distributions normal is always either the best
fit or the second best fit. Well use normal
distributions with means of zero and standard
deviations estimated from our sample data.
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It
looks like there is about a 5 chance of ending
up with less than 2.0500
million, so the 5 value at risk is

VaR

2.
3377
million
-
2.0500 million


0.2877 million


2
87
,
7
00


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Summary

• Advanced Simulation Applications
• Queueing Models
• Extend
• HOM
• Retirement Planning
• Market Shares
• Securities Pricing
• Asian Option
• Evaluation of Hedging Strategies
• Currency Risk